And, yes, they draw from a pool that eventually runs out. At that point, the only way to get new cards is to find merchants who sell them and/or play characters that give cards in various other situations (mostly gwent-focused quests but at least one quest has nothing to do with gwent but rewards you with a card anyway).
Awesome, that's what I needed to know. I was worried it was a bug, because for so long I'd received a card when I beat them the first time, but now I'm just getting crowns when I beat them the first time. Good to know it's not a bug.